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Theorem List for Metamath Proof Explorer - 11001-11100   *Has distinct variable group(s)
TypeLabelDescription
Statement

5.4.4  Some properties of specific numbers

This includes adding two pairs of values 1..10 (where the right is less than the left) and where the left is less than the right for the values 1..10.

Theoremneg1cn 11001 -1 is a complex number (common case). (Contributed by David A. Wheeler, 7-Jul-2016.)
-1 ∈ ℂ

Theoremneg1rr 11002 -1 is a real number (common case). (Contributed by David A. Wheeler, 5-Dec-2018.)
-1 ∈ ℝ

Theoremneg1ne0 11003 -1 is nonzero (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 ≠ 0

Theoremneg1lt0 11004 -1 is less than 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
-1 < 0

Theoremnegneg1e1 11005 --1 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
--1 = 1

Theorem1pneg1e0 11006 1 + -1 is 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + -1) = 0

Theorem0m0e0 11007 0 minus 0 equals 0 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(0 − 0) = 0

Theorem1m0e1 11008 1 - 0 = 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 − 0) = 1

Theorem0p1e1 11009 0 + 1 = 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(0 + 1) = 1

Theorem1p0e1 11010 1 + 0 = 1. (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 0) = 1

Theorem1p1e2 11011 1 + 1 = 2. (Contributed by NM, 1-Apr-2008.)
(1 + 1) = 2

Theorem2m1e1 11012 2 - 1 = 1. The result is on the right-hand-side to be consistent with similar proofs like 4p4e8 11041. (Contributed by David A. Wheeler, 4-Jan-2017.)
(2 − 1) = 1

Theorem1e2m1 11013 1 = 2 - 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
1 = (2 − 1)

Theorem3m1e2 11014 3 - 1 = 2. (Contributed by FL, 17-Oct-2010.) (Revised by NM, 10-Dec-2017.) (Proof shortened by AV, 6-Sep-2021.)
(3 − 1) = 2

Theorem4m1e3 11015 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.)
(4 − 1) = 3

Theorem5m1e4 11016 5 - 1 = 4. (Contributed by AV, 6-Sep-2021.)
(5 − 1) = 4

Theorem6m1e5 11017 6 - 1 = 5. (Contributed by AV, 6-Sep-2021.)
(6 − 1) = 5

Theorem7m1e6 11018 7 - 1 = 6. (Contributed by AV, 6-Sep-2021.)
(7 − 1) = 6

Theorem8m1e7 11019 8 - 1 = 7. (Contributed by AV, 6-Sep-2021.)
(8 − 1) = 7

Theorem9m1e8 11020 9 - 1 = 8. (Contributed by AV, 6-Sep-2021.)
(9 − 1) = 8

Theorem2p2e4 11021 Two plus two equals four. For more information, see "2+2=4 Trivia" on the Metamath Proof Explorer Home Page: mmset.html#trivia. This proof is simple, but it depends on many other proof steps because 2 and 4 are complex numbers and thus it depends on our construction of complex numbers. The proof o2p2e4 7508 is similar but proves 2 + 2 = 4 using ordinal natural numbers (finite integers starting at 0), so that proof depends on fewer intermediate steps. (Contributed by NM, 27-May-1999.)
(2 + 2) = 4

Theorem2times 11022 Two times a number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.) (Proof shortened by AV, 26-Feb-2020.)
(𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴))

Theoremtimes2 11023 A number times 2. (Contributed by NM, 16-Oct-2007.)
(𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴))

Theorem2timesi 11024 Two times a number. (Contributed by NM, 1-Aug-1999.)
𝐴 ∈ ℂ       (2 · 𝐴) = (𝐴 + 𝐴)

Theoremtimes2i 11025 A number times 2. (Contributed by NM, 11-May-2004.)
𝐴 ∈ ℂ       (𝐴 · 2) = (𝐴 + 𝐴)

Theorem2txmxeqx 11026 Two times a complex number minus the number itself results in the number itself. (Contributed by Alexander van der Vekens, 8-Jun-2018.)
(𝑋 ∈ ℂ → ((2 · 𝑋) − 𝑋) = 𝑋)

Theorem2div2e1 11027 2 divided by 2 is 1 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 / 2) = 1

Theorem2p1e3 11028 2 + 1 = 3. (Contributed by Mario Carneiro, 18-Apr-2015.)
(2 + 1) = 3

Theorem1p2e3 11029 1 + 2 = 3 (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)
(1 + 2) = 3

Theorem3p1e4 11030 3 + 1 = 4. (Contributed by Mario Carneiro, 18-Apr-2015.)
(3 + 1) = 4

Theorem4p1e5 11031 4 + 1 = 5. (Contributed by Mario Carneiro, 18-Apr-2015.)
(4 + 1) = 5

Theorem5p1e6 11032 5 + 1 = 6. (Contributed by Mario Carneiro, 18-Apr-2015.)
(5 + 1) = 6

Theorem6p1e7 11033 6 + 1 = 7. (Contributed by Mario Carneiro, 18-Apr-2015.)
(6 + 1) = 7

Theorem7p1e8 11034 7 + 1 = 8. (Contributed by Mario Carneiro, 18-Apr-2015.)
(7 + 1) = 8

Theorem8p1e9 11035 8 + 1 = 9. (Contributed by Mario Carneiro, 18-Apr-2015.)
(8 + 1) = 9

Theorem9p1e10OLD 11036 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) Obsolete version of 9p1e10 11372 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(9 + 1) = 10

Theorem3p2e5 11037 3 + 2 = 5. (Contributed by NM, 11-May-2004.)
(3 + 2) = 5

Theorem3p3e6 11038 3 + 3 = 6. (Contributed by NM, 11-May-2004.)
(3 + 3) = 6

Theorem4p2e6 11039 4 + 2 = 6. (Contributed by NM, 11-May-2004.)
(4 + 2) = 6

Theorem4p3e7 11040 4 + 3 = 7. (Contributed by NM, 11-May-2004.)
(4 + 3) = 7

Theorem4p4e8 11041 4 + 4 = 8. (Contributed by NM, 11-May-2004.)
(4 + 4) = 8

Theorem5p2e7 11042 5 + 2 = 7. (Contributed by NM, 11-May-2004.)
(5 + 2) = 7

Theorem5p3e8 11043 5 + 3 = 8. (Contributed by NM, 11-May-2004.)
(5 + 3) = 8

Theorem5p4e9 11044 5 + 4 = 9. (Contributed by NM, 11-May-2004.)
(5 + 4) = 9

Theorem5p5e10OLD 11045 5 + 5 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 5p5e10 11472 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(5 + 5) = 10

Theorem6p2e8 11046 6 + 2 = 8. (Contributed by NM, 11-May-2004.)
(6 + 2) = 8

Theorem6p3e9 11047 6 + 3 = 9. (Contributed by NM, 11-May-2004.)
(6 + 3) = 9

Theorem6p4e10OLD 11048 6 + 4 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 6p4e10 11474 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(6 + 4) = 10

Theorem7p2e9 11049 7 + 2 = 9. (Contributed by NM, 11-May-2004.)
(7 + 2) = 9

Theorem7p3e10OLD 11050 7 + 3 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 7p3e10 11479 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(7 + 3) = 10

Theorem8p2e10OLD 11051 8 + 2 = 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 8p2e10 11486 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(8 + 2) = 10

Theorem1t1e1 11052 1 times 1 equals 1. (Contributed by David A. Wheeler, 7-Jul-2016.)
(1 · 1) = 1

Theorem2t1e2 11053 2 times 1 equals 2. (Contributed by David A. Wheeler, 6-Dec-2018.)
(2 · 1) = 2

Theorem2t2e4 11054 2 times 2 equals 4. (Contributed by NM, 1-Aug-1999.)
(2 · 2) = 4

Theorem3t1e3 11055 3 times 1 equals 3. (Contributed by David A. Wheeler, 8-Dec-2018.)
(3 · 1) = 3

Theorem3t2e6 11056 3 times 2 equals 6. (Contributed by NM, 2-Aug-2004.)
(3 · 2) = 6

Theorem3t3e9 11057 3 times 3 equals 9. (Contributed by NM, 11-May-2004.)
(3 · 3) = 9

Theorem4t2e8 11058 4 times 2 equals 8. (Contributed by NM, 2-Aug-2004.)
(4 · 2) = 8

Theorem5t2e10OLD 11059 5 times 2 equals 10. (Contributed by NM, 5-Feb-2007.) Obsolete version of 5t2e10 11510 as of 8-Sep-2021. (New usage is discouraged.) (Proof modification is discouraged.)
(5 · 2) = 10

Theorem2t0e0 11060 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.)
(2 · 0) = 0

Theorem4d2e2 11061 One half of four is two. (Contributed by NM, 3-Sep-1999.)
(4 / 2) = 2

Theorem2nn 11062 2 is a positive integer. (Contributed by NM, 20-Aug-2001.)
2 ∈ ℕ

Theorem3nn 11063 3 is a positive integer. (Contributed by NM, 8-Jan-2006.)
3 ∈ ℕ

Theorem4nn 11064 4 is a positive integer. (Contributed by NM, 8-Jan-2006.)
4 ∈ ℕ

Theorem5nn 11065 5 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 ∈ ℕ

Theorem6nn 11066 6 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 ∈ ℕ

Theorem7nn 11067 7 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 ∈ ℕ

Theorem8nn 11068 8 is a positive integer. (Contributed by Mario Carneiro, 15-Sep-2013.)
8 ∈ ℕ

Theorem9nn 11069 9 is a positive integer. (Contributed by NM, 21-Oct-2012.)
9 ∈ ℕ

Theorem10nnOLD 11070 Obsolete version of 10nn 11390 as of 6-Sep-2021. (Contributed by NM, 8-Nov-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
10 ∈ ℕ

Theorem1lt2 11071 1 is less than 2. (Contributed by NM, 24-Feb-2005.)
1 < 2

Theorem2lt3 11072 2 is less than 3. (Contributed by NM, 26-Sep-2010.)
2 < 3

Theorem1lt3 11073 1 is less than 3. (Contributed by NM, 26-Sep-2010.)
1 < 3

Theorem3lt4 11074 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 4

Theorem2lt4 11075 2 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 4

Theorem1lt4 11076 1 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 4

Theorem4lt5 11077 4 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 5

Theorem3lt5 11078 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 5

Theorem2lt5 11079 2 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 5

Theorem1lt5 11080 1 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 5

Theorem5lt6 11081 5 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 6

Theorem4lt6 11082 4 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 6

Theorem3lt6 11083 3 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 6

Theorem2lt6 11084 2 is less than 6. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 6

Theorem1lt6 11085 1 is less than 6. (Contributed by NM, 19-Oct-2012.)
1 < 6

Theorem6lt7 11086 6 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 7

Theorem5lt7 11087 5 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 7

Theorem4lt7 11088 4 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 7

Theorem3lt7 11089 3 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 7

Theorem2lt7 11090 2 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 7

Theorem1lt7 11091 1 is less than 7. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 7

Theorem7lt8 11092 7 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
7 < 8

Theorem6lt8 11093 6 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
6 < 8

Theorem5lt8 11094 5 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
5 < 8

Theorem4lt8 11095 4 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
4 < 8

Theorem3lt8 11096 3 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
3 < 8

Theorem2lt8 11097 2 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
2 < 8

Theorem1lt8 11098 1 is less than 8. (Contributed by Mario Carneiro, 15-Sep-2013.)
1 < 8

Theorem8lt9 11099 8 is less than 9. (Contributed by Mario Carneiro, 19-Feb-2014.)
8 < 9

Theorem7lt9 11100 7 is less than 9. (Contributed by Mario Carneiro, 9-Mar-2015.)
7 < 9

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